arXiv subjects

Mondher Chouikhi

Publications and source records attributed to Mondher Chouikhi.

5 recordsLinked to original sources

arXiv2025

Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

In this paper, we give a geometric approach to the cubic oscillator with three distinct turning points based on the $\mathcal{D\diagup SG}$\emph{\ correspondence }introduced in \cite{Thabet+al}. The existence of quantization conditions, depending on extra data for the potential, is related to some particular critical graphs of the quadratic differential $λ^{2}\left(z-a\right) \left( z^{2}-1\right) dz^{2}$ where $λ$ is a non vanishing complex number, $a\in \mathbb{C}\diagdown \left\{ -1,1\right\}$. We investigate this geometric approach in two level: the first level is studying an inverse spectral problem related to cubic oscillator. The second level describes the zeros locations of eigenfunctions related to this oscillator. Our results may provide a geometric proof of some questions related to cubic potential case.

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arXiv2024

Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories

In this paper, we give a full description of the critical graph of the quadratic differential $\varpi_{a,θ}$ defined on the Riemann sphere $\widehat{% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion }$ by $\varpi_{a,θ}=-e^{2iθ}\left( z-a\right) \left( z^{2}-1\right) dz^{2},$ where $θ\in% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ and $a\in% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion .$. We prove that the existence and the number of short trajectories of $\varpi_{a,θ}$ depend on the location of $a$ in certain curves defined on the complex plane as the level sets of some harmonic functions. More focus will be to the cases $θ\in\left\{ 0,π/4\right\} .$ We investigate these classifications to study an inverse spectral problem related to the complex cubic oscillator for Schrödinger equation.

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arXiv2017

Trajectories of a quadratic differential related to a quasi-exactly solvable sextic oscillator

In this paper, we discuss the existence of solution (as Cauchy transform of a signed measure) of a particular algebraic quadratic equation of the form $\mathcal{C}^{2}\left( z\right) +r\left( z\right) \mathcal{C}\left( z\right) +s\left( z\right) =0.$ This problem remains to describe the critical graph of a related polynomial quadratic differential; in particular, we discuss the existence of finite critical trajectories of this quadratic differential.

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arXiv2017

Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential

In this paper, we study the weak asymptotic in the plane of some wave functions resulting from the WKB techniques applied to a Shrodinger equation with quartic oscillator and having some boundary condition. In first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions .Especially , we investigate the properties of the Cauchy transform of the root counting measure of a re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form $\mathcal{C}^{2}\left( z\right) +r\left( z\right) \mathcal{C}\left( z\right) +s\left( z\right) =0$, where $r,s$ are also polynomials. In second step, we discuss the existence of solutions (as Cauchy transform of a signed measures) of this algebraic equation.This problem remains to describe the critical graph of a related 4-degree polynomial quadratic differential $-p\left( z\right) dz^{2}$. In particular, we discuss the existence(and their number) of finite critical trajectories of this quadratic differential.

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arXiv2015

On the existence of short trajectories of quadratic differentials related to generalized Jacobi polynomials with non real varying parameters

The study of the asymptotic distributions of zeros of generalized Jacobi polynomials with non real varying parameters, leads with quadratic differentials. In fact, the support of the limit measure of the root-counting measures sits on the finite critical trajectories of a related quadratic differential. In this paper, we study the trajectories of this quadratic differential, more precisely, we give a necessary and sufficient condition on the complex numbers a; b; and λ for the existence of at list one finite critical trajectory of the quadratic differential ((λ^2(z-a)(z-b))/((z^2-1)^2))dz^2.

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